In this work, it is proved that it makes sense to pursue the idea of de veloping a theory of surreal super p-branes. One uses the fact that the real numbers are contained in the surreal number set and therefore at the classical level all functions in the string action, or in general in the p-brane action, can be considered as quantities build in terms of the fundamental blocks of dyadic rational fractions. However, a partial basic formalism is only presented. Therefore, this work can be considered as a progress report, which, in a sense, may motivate, among other things, the introduction of an approach in which relativistic tops and p-branes are combined.